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Theorem tgptmd 21025
Description: A topological group is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
Assertion
Ref Expression
tgptmd  |-  ( G  e.  TopGrp  ->  G  e. TopMnd )

Proof of Theorem tgptmd
StepHypRef Expression
1 eqid 2429 . . 3  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
2 eqid 2429 . . 3  |-  ( invg `  G )  =  ( invg `  G )
31, 2istgp 21023 . 2  |-  ( G  e.  TopGrp 
<->  ( G  e.  Grp  /\  G  e. TopMnd  /\  ( invg `  G )  e.  ( ( TopOpen `  G )  Cn  ( TopOpen
`  G ) ) ) )
43simp2bi 1021 1  |-  ( G  e.  TopGrp  ->  G  e. TopMnd )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1870   ` cfv 5601  (class class class)co 6305   TopOpenctopn 15279   Grpcgrp 16620   invgcminusg 16621    Cn ccn 20171  TopMndctmd 21016   TopGrpctgp 21017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-nul 4556
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-iota 5565  df-fv 5609  df-ov 6308  df-tgp 21019
This theorem is referenced by:  tgptps  21026  tgpcn  21030  tgpsubcn  21036  tgpmulg  21039  oppgtgp  21044  tgplacthmeo  21049  subgtgp  21051  clsnsg  21055  tgpt0  21064  prdstgpd  21070  tsmssub  21094  tsmsxp  21100  trgtmd2  21114  nlmtlm  21627  qqhcn  28634
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